Probability of Drawing the Same Ordered Set of Three Cards Twice from a Deck of 72 Cards

When it comes to playing or analyzing games involving cards, understanding the probability of specific events is essential. This article delves into the probability of drawing the same ordered set of three cards twice from a deck of 72 cards. We will explore the mathematical calculations and practical applications involved in such a scenario.

Calculating the Probability

To calculate the probability of drawing the same set of three cards in order from a deck of 72 cards, let's break down the steps:

Total Number of Ordered Combinations: For the first card, there are 72 options. For the second card, there are 71 options since one card has already been drawn. For the third card, there are 70 options.

Therefore, the total number of ordered combinations is:

72 x 71 x 70 343,200

Probability of Drawing the Same Three Cards in the Same Order Again:

Once the first three cards have been drawn, only one specific sequence of three cards matches that order. The remaining deck has 69 cards left, and we need to calculate the probability of drawing these three cards in the same order:

1 / (69 x 68 x 67) 1 / 346,344

Combined Probability:

The probability of drawing the same three cards in order twice can be expressed as:

1 / (72 x 71 x 70) x 1 / (69 x 68 x 67) 1 / 119,800,832,000

This gives us the final probability of drawing the same set of three cards in order twice.

Conclusion

The probability of the same ordered set of three cards being drawn twice in succession from a deck of 72 cards is extremely low. This is due to the specific sequence required among a large number of possibilities.

Practical Application: Tarot Readings

While the 72-card deck is less common, let's explore how this concept applies to a 78-card tarot deck. In tarot readings, the order of the cards usually matters, and each card can be either upright or inverted, leading to permutations instead of combinations.

Initial Calculation:

For a 78-card deck, with each card having two orientations:

78 x 2 x 77 x 2 x 76 x 2 3,651,648

Three-Card Spread:

Commonly, the first card represents the past, the second the present, and the third the future. Additional elaborating cards may be drawn for each position, especially for the present and future:

78 x 2 x 77 x 2 x 76 x 2 x 75 x 2 x 74 x 2 8,106,658,560,000

These numbers illustrate the vast number of possible three-card tarot spreads, even with elaborating cards.

Final Thoughts

While the probability of drawing the same ordered set of three cards twice from a 72-card deck is exceedingly low, the rich and intricate world of tarot readings showcases the complexity and potential of card combinations and sequences. Understanding these concepts can enhance one's appreciation of card games and gradually applied fields such as tarot readings.

Keywords: deck of cards, ordered combinations, probability calculation